Log[z] == ((Sqrt[-1 - I z] Sqrt[I z - 1])/Sqrt[1 + z^2]) ((Sqrt[z^2]/z) ArcCsch[(2 z)/(z^2 - 1)] + (Pi (z - Sqrt[z^2]))/ (2 Sqrt[-z^2])) + ((Pi I)/2) (Sqrt[z/(z + I)] Sqrt[(z + I)/z] - Sqrt[z/(z - I)] Sqrt[(z - I)/z])

Cell[BoxData[RowBox[List[RowBox[List["Log", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "-", "1"]]]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[SqrtBox[SuperscriptBox["z", "2"]], "z"], " ", RowBox[List["ArcCsch", "[", FractionBox[RowBox[List["2", " ", "z"]], RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["z", "-", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox["z", RowBox[List["z", "+", "\[ImaginaryI]"]]]], " ", SqrtBox[FractionBox[RowBox[List["z", "+", "\[ImaginaryI]"]], "z"]]]], "-", RowBox[List[SqrtBox[FractionBox["z", RowBox[List["z", "-", "\[ImaginaryI]"]]]], " ", SqrtBox[FractionBox[RowBox[List["z", "-", "\[ImaginaryI]"]], "z"]]]]]], ")"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mi> &#8520; </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> <mo> - </mo> <mrow> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> - </mo> <mi> &#8520; </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mi> z </mi> </mfrac> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ln /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <imaginaryi /> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccsch /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Log", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["ArcCsch", "[", FractionBox[RowBox[List["2", " ", "z"]], RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], "]"]]]], "z"], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["z", "-", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]]]], ")"]]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox["z", RowBox[List["z", "+", "\[ImaginaryI]"]]]], " ", SqrtBox[FractionBox[RowBox[List["z", "+", "\[ImaginaryI]"]], "z"]]]], "-", RowBox[List[SqrtBox[FractionBox["z", RowBox[List["z", "-", "\[ImaginaryI]"]]]], " ", SqrtBox[FractionBox[RowBox[List["z", "-", "\[ImaginaryI]"]], "z"]]]]]], ")"]]]]]]]]]]

2001-10-29